introduction to condensed matter · pdf filediatomic linear chain ... normal modes. we may...

M.P. Vaughan

Introduction to Condensed

Matter Physics

Phonons II

Overview

Coláiste na hOllscoile Corcaigh, Éire

University College Cork, Ireland ROINN NA FISICE Department of Physics

PY3105 Introduction to

Condensed Matter Physics

Overview of phonons II

• Monatomic linear chain

• The harmonic potential

• 3D monatomic crystal

• The dynamical matrix

• 3D crystal with a basis

• Measuring dispersion relations

• Macroscopic properties

The Monatomic Linear

Chain





Diatomic linear chain

In the previous lecture, we analysed the monatomic linear

chain in terms of classical modes.






This yielded two coupled differential equations with

eigenvalues of the angular frequency

,

cos1211

2/1

22

2

2

1

2

2

2

12

2

2

1

2

qa

where

.and2

2

2

1

2

1m

k

m

k SS






The wavevectors q were given by

where N is the number of links (containing 2 atoms) in the

chain.

,2

N

n

aq

Plotting the eigenvalues against wavevector, we obtained

graphs illustrating two phonon branches:

1. acoustic phonons

2. optical phonons





Dispersion relations

Optical modes

Acoustic modes

Linear approximation for

acoustic branch





Monatomic linear chain

The solution for the monatomic linear chain is less

involved. For completeness, we give it here.

In this case, we have just one differential equation

.11 nnSnnSn uukuukum

Looking for solutions of the form

,tqnai

n Aeu

we have

.112 iqa

S

iqa

S ekekm





Monatomic linear chain

Writing this in terms of

we have

.cos12 2

1

2 qa

,2

1m

kS

Note that this is only an acoustic mode solution – there are

no optical modes.

The Harmonic Potential





The harmonic potential

Limiting the expansion of the potential energy of the ring to

terms quadratic in un, the potential energy of the monatomic

ring may be written

.2

2

1

2

10 n

nnnnS uuuu

kUU

Differentiating this with respect to un gives

.11

nnSnnS

n

uukuuku

U






Multiplying by -1 then just gives the force on the mass at

position n, so

which reproduces the differential equation solved earlier.

,11

nnSnnS

n

n uukuuku

Uum

The same strategy may be applied to the diatomic linear

chain. In this case, the harmonic potential is

.2

2

,2,1

2

1,2,10

n

nnnnS uuuu

kUU






.,2,11,2,1

,1

nnSnnS

n

uukuuku

U

Differentiating this with respect to u1,n gives

Meanwhile, differentiating this with respect to u2,n gives

.,1,21,1,2

,2

nnSnnS

n

uukuuku

U

(Tip: consider the n+1 terms in the summation in U).






n

nu

Uum

,1

,11

Hence, putting

and

,,2

,22

n

nu

Uum

reproduces the coupled equations

.

,

,21,1,1,2,22

,1,21,2,1,11

nnSnnSn

nnSnnSn

uukuukum

uukuukum






Let us consider an alternative form for the harmonic potential

of a monatomic chain. Putting U0 = 0, we can write the

contribution to U due to a single link at position n as

.2

111 nnSnnn uukuuU

Putting U0 = 0 and extending the summation beyond nearest

neighbour interactions, we have

nn

nnnnnn uukuuU'

'''4

1

(the additional factor of ½ ensures that we do not double

count distinct links).






The total potential is usually written in the form

.2

1

'

''nn

nnnn uDuU

.4

1

4

1

'

''

'

' nn

nnnn

nn

nnnn ukuukuU

,'' nnnn kk To justify this we note that, since we can write the

potential as






.'''

''',' nn

n

nnnnnn kkD

We now define

(Note that the second term is not included in the

summation). Inserting this into the expression for U

.4

1

,4

1

'

'''

''

''',

'

'''

nn

nnnnn

n

nnnnn

nn

nnnnnn

ukuuku

uukuuU






,4

1

4

1

,4

1

4

1

'

''

'

'

'

''

''

''

nn

nnnn

nn

nnnn

nn

nnnn

n

n

n

nnn

ukuuku

ukuukuU

Separating this into two multiple summations and employing

the Kronecker delta,

which reproduces the original form of the potential.

The utility of the new form for the potential becomes

apparent when we move to 3D crystals.





The 3D monatomic crystal

Generalising to 3D, the displacements may now be

expressed as vectors u associated with a lattice point Rn.

,2

1

'

''nn

nnn

T

nU uDu

The potential may now be written in matrix form as

where Dnn’ is a matrix and the T superscript denotes the

operation of transposition.

We shall not attempt to solve this system exactly here, just

highlight features of the general solution.






We now require solutions of the form

where n is the polarisation vector giving the direction of the

ionic displacements.

The equations of motion for a monatomic crystal are

,

ti

nne

Rq

nu

,,

,

n

nu

Uum

where labels the component of un.





The dynamical matrix

From the expression for U, we have

Hence

.'

''

,

n

nnn

nu

UuD

is known as the dynamical matrix.

,2nDn qm

where

'

''

n

i

nnne

Rq

q DD






For a monatomic crystal of N atoms, we would then have 3N

normal modes. We may obtain these from the 3 couled

differential equations.

As in the case of the monatomic linear chain, the 3D

monatomic crystal yields only acoustic modes.

These modes may be differentiated on the basis of

polarisation. We have

1. One longitudinal mode (LA)

2. Two transverse modes (TA), orthogonal to the

direction of propagation and to each other





The 3D crystal with a basis

Restricting our consideration of the number of modes to a

particular value of q, the number of modes for the 3D

monatomic crystal is 3.

For a 3D crystal with a basis of p atoms, the number of

modes per q is 3p. These are categorised as follows

• 1 longitudinal acoustic (LA) mode

• 2 transverse acoustic (TA) modes

• p -1 longitudinal optical (LO) modes

• 2(p -1) transverse optical (TO) modes

Measuring dispersion

relations





Measuring dispersion relations (1)

• Neutron Inelastic Scattering

• Due to penetrative power of neutrons, can

be used to measure bulk phonons

• Electron Energy Loss Spectroscopy

(EELS)

• Uses beam of electrons with narrow range

of energies. Used for surface phonons.





Measuring dispersion relations (2)

• Inelastic Atomic Scattering

• Due to low penetrability of (He) atoms,

limited to surface phonons

• Raman Scattering

• Inelastic scattering of light due to

absorption (sometimes emission) of energy

by optical modes.





Brillouin zone of diamond / zinc blende





Diamond





Silicon





GaAs

Macroscopic properties





Macroscopic properties

• The properties of phonons are

related to the macroscopic properties

of a material. A few of these are

• Elastic properties

• Thermal conductivity

• Electrical resistance

introduction to condensed matter · pdf filediatomic linear chain ... normal modes. we may...

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